reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;

theorem Th51:
  f is dilatation & f.a=a & f.b=b & a<>b implies f=(id the carrier of OAS)
proof
  assume that
A1: f is dilatation and
A2: f.a=a and
A3: f.b=b and
A4: a<>b;
  now
    let x;
A5: now
      assume
A6:   a,b,x are_collinear;
      now
        consider z such that
A7:     not a,b,z are_collinear by A4,DIRAF:37;
        assume
A8:     x<>a;
A9:     not a,z,x are_collinear
        proof
          assume a,z,x are_collinear;
          then
A10:      a,x,z are_collinear by DIRAF:30;
          a,x,a are_collinear & a,x,b are_collinear by A6,DIRAF:30,31;
          hence contradiction by A8,A7,A10,DIRAF:32;
        end;
        f.z=z by A1,A2,A3,A7,Th50;
        hence f.x=x by A1,A2,A9,Th50;
      end;
      hence f.x=x by A2;
    end;
    not a,b,x are_collinear implies f.x=x by A1,A2,A3,Th50;
    hence f.x=(id the carrier of OAS).x by A5;
  end;
  hence thesis by FUNCT_2:63;
end;
